How to render the Trajectory of a Moving Object?
You can use dynamic paint, but as far as I know, this will alias a bit.



So I propose to use Animation Nodes instead. For 2.8 you can find an experimental version here.



The principle is to create a Bezier curve, set it to 2D, bevel it a bit and use it in Animation Nodes.



Calculation steps:


Take the moving sphere, its location and append it to the spline so that it is reinjected in the curve (bottom line of the nodes).
The used spline (top part of the nodes) is either an empty one (at frame 1, so that the curve starts to be empty) or the current spline of the curve as it is.


Few words about the cycloide itself.

If you add a simple deform (bend mode) to a subdivided segment, this will make a circle:



If we add an empty as axis origin, then moving the empty will make the obtained circle to rotate.

To make this rotation visible, add another empty (below with a sphere shape), and make the 3 first vertices of the segment parent of this empty.

Now the empty will move and rotate following the segment made circular thanks to the simple deform modifier:



The "drawing sphere above" is simply parented to this sphere empty.



(the file contains also the Dynamic Paint setting, using vertex color, it is on the hidden plane)